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Theorem tncp 20861
Description: There exist three non colinear points. (Contributed by FL, 3-Aug-2009.)
Hypothesis
Ref Expression
tncp.1  |-  P  = 
U. L
Assertion
Ref Expression
tncp  |-  ( L  e.  Plig  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  A. l  e.  L  -.  (
a  e.  l  /\  b  e.  l  /\  c  e.  l )
)
Distinct variable groups:    L, a,
b, c, l    P, a, b, c
Allowed substitution hint:    P( l)

Proof of Theorem tncp
StepHypRef Expression
1 tncp.1 . . . 4  |-  P  = 
U. L
21isplig 20860 . . 3  |-  ( L  e.  Plig  ->  ( L  e.  Plig  <->  ( A. a  e.  P  A. b  e.  P  ( a  =/=  b  ->  E! l  e.  L  ( a  e.  l  /\  b  e.  l ) )  /\  A. l  e.  L  E. a  e.  P  E. b  e.  P  (
a  =/=  b  /\  a  e.  l  /\  b  e.  l )  /\  E. a  e.  P  E. b  e.  P  E. c  e.  P  A. l  e.  L  -.  ( a  e.  l  /\  b  e.  l  /\  c  e.  l ) ) ) )
32ibi 232 . 2  |-  ( L  e.  Plig  ->  ( A. a  e.  P  A. b  e.  P  (
a  =/=  b  ->  E! l  e.  L  ( a  e.  l  /\  b  e.  l ) )  /\  A. l  e.  L  E. a  e.  P  E. b  e.  P  (
a  =/=  b  /\  a  e.  l  /\  b  e.  l )  /\  E. a  e.  P  E. b  e.  P  E. c  e.  P  A. l  e.  L  -.  ( a  e.  l  /\  b  e.  l  /\  c  e.  l ) ) )
43simp3d 969 1  |-  ( L  e.  Plig  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  A. l  e.  L  -.  (
a  e.  l  /\  b  e.  l  /\  c  e.  l )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   E!wreu 2558   U.cuni 3843   Pligcplig 20858
This theorem is referenced by:  lpni  20862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563  df-v 2803  df-uni 3844  df-plig 20859
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